The density dependence of the transition temperature in a homogenous Bose fluid J.D. Reppy, B.C. Crookera, B. Hebralb, A.D. Corwin, J. He, and G.M. Zassenhausc Laboratory of Atomic and Solid State Physics and The Cornell Center for Materials Research, Clark Hall, Cornell University, Ithaca, NY 14853-2501 (November 5, 2018) Transition temperature data obtained as a function of particle density in the 4He-Vycor system are compared with recent theoretical calculations for 3D Bose condensed systems. In the low density dilute Bose gas regime we find, in agreement with theory, a positive shift in the transition tempera- 3 1/3 ture of the form ∆T/T0 = γ(na ) . At higher densities a maximum is found in the ratio of Tc/T0 for a value of the interaction parameter, na3, that is in agreement with path-integral Monte Carlo calculations. PACS numbers: 03.75.Fi, 05.30.Jp, 67.40.-w The role of interparticle interactions in the determina- inaccessible with the alkali vapor systems. Further, work- tion of the properties of low density Bose-Einstein con- ing with the 4He-Vycor system allows much larger sample densed (BEC) systems has been a topic of interest for sizes, on the order a cm3, with an essentially unlimited many years. In spite of a long history of theoretical time for observation of the BEC superfluid state. investigation [1] dating back to the 1950s, elementary In this letter we will give only a brief summary of our questions such as the possible shift in the transition tem- experimental methods. The cryogenic techniques and perature, Tc, with density and interaction strength have thermometry methods are those standard in the study of remained unsettled until the recent past. In the case of superfluid 3He, and the reader is referred elsewhere [10] the repulsive interactions in the dilute Bose gas, there for details. For the present experiments, we have used a has now emerged a consensus [2] [3] [4] that Tc will be torsional oscillator technique to obtain a signal propor- an increasing function of the interaction parameter, na3, tional to the superfluid particle density within our Vycor where a is the hard sphere diameter and n the particle sample, which then allows us to estimate the number den- density. This may seem a surprising result, since it is sity for the particles participating in the superfluid and well known that in the case of liquid 4He the superfluid as well as to determine the superfluid transition temper- transition occurs at a temperature well below the transi- ature. The interior channels of the porous Vycor glass tion temperature, T0, for an ideal Bose gas with the same used for these measurements range in diameter from 4 particle mass and number density. Moreover, a number to 8 nm and form a highly interconnected 3D network. of the earlier calculations [5] had found that the transi- The superfluid helium atoms are constrained by van der tion temperature would be reduced as a consequence of Waals forces to move over the complex 3D-connected sur- interparticle interaction. face provided by the pores. It is important to appreciate Motivated by the recent theoretical work in this area, that at low temperatures the thermal wavelength of the we have examined the dependence of the transition tem- mobile superfluid particles can be larger than the pore perature on superfluid particle number for the 4He-Vycor size and that the Feynman exchange cycles characteriz- system. In our early work with this system we demon- ing the BEC or superfluid state will link many pores at strated, for the first time in 1983 [6], an experimental re- low particle density [11]. Therefore we model the su- alization of the weakly interacting or “dilute” Bose gas. perfluid phase as a homogeneous Bose gas constrained arXiv:cond-mat/9910013v1 1 Oct 1999 The lowest density achieved in these experiments was on within the volume of the Vycor sample. We anticipate, the order of 2 × 1018 per cm3. This is sufficiently low to however, that the influence of the substrate and pore ge- provide a region of overlap, in terms of the interaction ometry may be reflected in a small modification of the parameter, with the values of na3, currently accessible effective mass of the mobile superfluid atoms. We then to the BEC systems realized with 87Rb [7] or 23Na [8] expect to observe, in the low density limit, thermody- atoms confined within magnetic or optical traps. In the namic properties similar to those of the “free” or ideal case of Bose condensed atomic hydrogen [9] the small s- Bose gas with an effective mass, m∗. wave scattering length and limits on the particle density, In the discussion of our experimental results, we shall set by recombination, restrict this system to values of na3 first consider the data for the dilute Bose gas regime. The several orders of magnitude below the values that can be quantities required for a comparison with theory are the achieved with Bose-condensed Na, Rb or He. particle number density and the corresponding transition For questions such as the effect on Tc of increasing the temperature. The number density is estimated from an interaction parameter, the 4He-Vycor system offers ad- extrapolation of the superfluid signal to zero tempera- vantages over the BEC systems of trapped atomic gases, ture and the calibrated mass sensitivity of our torsional because in the Vycor case, the interaction parameter can oscillator. Following the recent calulations [2] [3] [4], we be varied continuously from the low-density, weakly inter- expect that for low densities Tc will be given by acting limit to the strongly interacting regime currently 1 3 1/3 Tc = T0[1 + γ(na ) ], (1) and result in a range of overlap in terms of the interaction parameter between all three systems. This is illustrated 2 ∗ 2/3 2/3 where T0 = [2π¯h /m kB ζ(3/2) ]n is the transition in Figure 1, where we indicate the range of the quan- ∗ temperature for an ideal Bose gas with particle mass m tity n1/3a for the 4He, experiments 23Na and 87Rb ex- and density n. The coefficient for γ is positive as required periments. Since the 4He-Vycor system clearly exhibits 3 for Tc to rise above T0 as the interaction parameter, na superfluid properties, one may also expect superfluidity increases. Although there is agreement on the form of to exists in the Bose-condensed alkali systems at compa- Eq. 1, the theoretical estimates for γ range over more rable values of the interaction parameter and sufficiently than an order of magnitude, from an estimate of 0.34 low temperature. reported by Gr¨uter, Ceperley, and Lalo¨e (GCL) [3], 2.9 by Baym et al. [4], to a value of 4.66 given by Stoof [2]. 0.05 Stoof For a convenient comparison to experimental results Best Fit 1/3 we cast Eq. 1 in a linear form with the variable, n a, 0.04 Baym et al. as 2 0.03 Ideal Gas Tc 2π¯h 1/3 ) = [1 + γ(n a)]. (2) 2/3 ∗ 2/3 k ( n m kBζ(3/2) c T 0.02 2/3 In Figure 1, we then plot Tc/n against the parameter n1/3a, taking a value of 0.22 nm [12] for the helium hard 0.01 core diameter. As expected from theory, a linear fit gives a good representation for our data in the low density 0.00 regime. The zero density intercept of the fit serves to 0 10 20 30 40 50 ∗ determine an effective mass ratio of m /m = 1.34, where n (1018 atoms/cc) m is the 4He mass, and the slope yields a value of 5.1 for γ, which is within a factor of 1.09 of the value given by FIG. 2. The observed transition temperature, Tc, is plot- Stoof. ted as a function of the particle density, n. The top curve through the data is the function given by Eq. 1 with the pa- ∗ 5 rameters for the effective mass, m and γ determined as from 3 / 2 ) the linear fit shown in Fig. 1. The two other curves are for γ 4He cc = 4.66 as given by Stoof and γ = o, the noninteracting case. / m 4 o t 1H(x10) In Figure 2 we show a more conventional plot of the a ( low density data. Here we have plotted T as a function / c K of the particle density, n. The curve through the data is 15 - 3 23 the theoretical expression given in Eq. 1 with our best Na ∗ 10 ( fit values for γ and m as determined previously. Two 87 3 / Rb 2 curves based on the γ estimates of Stoof and Baym et al. n / are also shown. The lowest curve is for the case of the c 2 T 0.00 0.02 0.04 0.06 0.08 noninteracting Bose gas case (i.e., γ = 0 in Eq. 1). The 1/3 relatively close agreement with the estimate of Stoof is n a clear in this plot. In contrast to the other theoretical treatments of the 2/3 FIG. 1. The quantity Tc/n is plotted as a function of BEC problem for the interacting Bose gas [2] [4], the 1 3 the parameter n / a. The straight line is our fit to these calculations of GCL are not restricted to the low den- data. The arrows indicate the range of interaction parameter sity limit. The path-integral Monte Carlo technique em- 1 4 87 23 covered in the H, He, Rb, and Na experiments.